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Mathematics > Probability

Title:
Percolation on stationary tessellations: models, mean values and second order structure

Abstract: We consider a stationary face-to-face tessellation $X$ of $\mathbb{R}^d$ and
introduce several percolation models by colouring some of the faces black in a
consistent way. Our main model is cell percolation, where cells are declared
black with probability $p$ and white otherwise. We are interested in geometric
properties of the union $Z$ of black faces. Under natural integrability
assumptions we first express asymptotic mean-values of intrinsic volumes in
terms of Palm expectations associated with the faces. In the second part of the
paper we study asymptotic covariances of intrinsic volumes of $Z\cap W$, where
the observation window $W$ is assumed to be a polytope. Here we need to assume
the existence of suitable asymptotic covariances of the face processes of $X$.
We check these assumptions in the important special case of a Poisson Voronoi
tessellation. In the case of cell percolation on a normal tessellation,
especially in the plane, our formulae simplify considerably.